3.410 \(\int x^{13} (a+b x^2)^{9/2} \, dx\)

Optimal. Leaf size=140 \[ \frac{15 a^2 \left (a+b x^2\right )^{19/2}}{19 b^7}-\frac{20 a^3 \left (a+b x^2\right )^{17/2}}{17 b^7}+\frac{a^4 \left (a+b x^2\right )^{15/2}}{b^7}-\frac{6 a^5 \left (a+b x^2\right )^{13/2}}{13 b^7}+\frac{a^6 \left (a+b x^2\right )^{11/2}}{11 b^7}+\frac{\left (a+b x^2\right )^{23/2}}{23 b^7}-\frac{2 a \left (a+b x^2\right )^{21/2}}{7 b^7} \]

[Out]

(a^6*(a + b*x^2)^(11/2))/(11*b^7) - (6*a^5*(a + b*x^2)^(13/2))/(13*b^7) + (a^4*(a + b*x^2)^(15/2))/b^7 - (20*a
^3*(a + b*x^2)^(17/2))/(17*b^7) + (15*a^2*(a + b*x^2)^(19/2))/(19*b^7) - (2*a*(a + b*x^2)^(21/2))/(7*b^7) + (a
+ b*x^2)^(23/2)/(23*b^7)

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Rubi [A]  time = 0.0797439, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac{15 a^2 \left (a+b x^2\right )^{19/2}}{19 b^7}-\frac{20 a^3 \left (a+b x^2\right )^{17/2}}{17 b^7}+\frac{a^4 \left (a+b x^2\right )^{15/2}}{b^7}-\frac{6 a^5 \left (a+b x^2\right )^{13/2}}{13 b^7}+\frac{a^6 \left (a+b x^2\right )^{11/2}}{11 b^7}+\frac{\left (a+b x^2\right )^{23/2}}{23 b^7}-\frac{2 a \left (a+b x^2\right )^{21/2}}{7 b^7} \]

Antiderivative was successfully verified.

[In]

Int[x^13*(a + b*x^2)^(9/2),x]

[Out]

(a^6*(a + b*x^2)^(11/2))/(11*b^7) - (6*a^5*(a + b*x^2)^(13/2))/(13*b^7) + (a^4*(a + b*x^2)^(15/2))/b^7 - (20*a
^3*(a + b*x^2)^(17/2))/(17*b^7) + (15*a^2*(a + b*x^2)^(19/2))/(19*b^7) - (2*a*(a + b*x^2)^(21/2))/(7*b^7) + (a
+ b*x^2)^(23/2)/(23*b^7)

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int x^{13} \left (a+b x^2\right )^{9/2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int x^6 (a+b x)^{9/2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{a^6 (a+b x)^{9/2}}{b^6}-\frac{6 a^5 (a+b x)^{11/2}}{b^6}+\frac{15 a^4 (a+b x)^{13/2}}{b^6}-\frac{20 a^3 (a+b x)^{15/2}}{b^6}+\frac{15 a^2 (a+b x)^{17/2}}{b^6}-\frac{6 a (a+b x)^{19/2}}{b^6}+\frac{(a+b x)^{21/2}}{b^6}\right ) \, dx,x,x^2\right )\\ &=\frac{a^6 \left (a+b x^2\right )^{11/2}}{11 b^7}-\frac{6 a^5 \left (a+b x^2\right )^{13/2}}{13 b^7}+\frac{a^4 \left (a+b x^2\right )^{15/2}}{b^7}-\frac{20 a^3 \left (a+b x^2\right )^{17/2}}{17 b^7}+\frac{15 a^2 \left (a+b x^2\right )^{19/2}}{19 b^7}-\frac{2 a \left (a+b x^2\right )^{21/2}}{7 b^7}+\frac{\left (a+b x^2\right )^{23/2}}{23 b^7}\\ \end{align*}

Mathematica [A]  time = 0.0441173, size = 83, normalized size = 0.59 \[ \frac{\left (a+b x^2\right )^{11/2} \left (97240 a^2 b^4 x^8-45760 a^3 b^3 x^6+18304 a^4 b^2 x^4-5632 a^5 b x^2+1024 a^6-184756 a b^5 x^{10}+323323 b^6 x^{12}\right )}{7436429 b^7} \]

Antiderivative was successfully verified.

[In]

Integrate[x^13*(a + b*x^2)^(9/2),x]

[Out]

((a + b*x^2)^(11/2)*(1024*a^6 - 5632*a^5*b*x^2 + 18304*a^4*b^2*x^4 - 45760*a^3*b^3*x^6 + 97240*a^2*b^4*x^8 - 1
84756*a*b^5*x^10 + 323323*b^6*x^12))/(7436429*b^7)

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Maple [A]  time = 0.005, size = 80, normalized size = 0.6 \begin{align*}{\frac{323323\,{x}^{12}{b}^{6}-184756\,a{x}^{10}{b}^{5}+97240\,{a}^{2}{x}^{8}{b}^{4}-45760\,{a}^{3}{x}^{6}{b}^{3}+18304\,{a}^{4}{x}^{4}{b}^{2}-5632\,{a}^{5}{x}^{2}b+1024\,{a}^{6}}{7436429\,{b}^{7}} \left ( b{x}^{2}+a \right ) ^{{\frac{11}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^13*(b*x^2+a)^(9/2),x)

[Out]

1/7436429*(b*x^2+a)^(11/2)*(323323*b^6*x^12-184756*a*b^5*x^10+97240*a^2*b^4*x^8-45760*a^3*b^3*x^6+18304*a^4*b^
2*x^4-5632*a^5*b*x^2+1024*a^6)/b^7

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^13*(b*x^2+a)^(9/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.72266, size = 352, normalized size = 2.51 \begin{align*} \frac{{\left (323323 \, b^{11} x^{22} + 1431859 \, a b^{10} x^{20} + 2406690 \, a^{2} b^{9} x^{18} + 1826110 \, a^{3} b^{8} x^{16} + 530959 \, a^{4} b^{7} x^{14} + 231 \, a^{5} b^{6} x^{12} - 252 \, a^{6} b^{5} x^{10} + 280 \, a^{7} b^{4} x^{8} - 320 \, a^{8} b^{3} x^{6} + 384 \, a^{9} b^{2} x^{4} - 512 \, a^{10} b x^{2} + 1024 \, a^{11}\right )} \sqrt{b x^{2} + a}}{7436429 \, b^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^13*(b*x^2+a)^(9/2),x, algorithm="fricas")

[Out]

1/7436429*(323323*b^11*x^22 + 1431859*a*b^10*x^20 + 2406690*a^2*b^9*x^18 + 1826110*a^3*b^8*x^16 + 530959*a^4*b
^7*x^14 + 231*a^5*b^6*x^12 - 252*a^6*b^5*x^10 + 280*a^7*b^4*x^8 - 320*a^8*b^3*x^6 + 384*a^9*b^2*x^4 - 512*a^10
*b*x^2 + 1024*a^11)*sqrt(b*x^2 + a)/b^7

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Sympy [A]  time = 80.1342, size = 277, normalized size = 1.98 \begin{align*} \begin{cases} \frac{1024 a^{11} \sqrt{a + b x^{2}}}{7436429 b^{7}} - \frac{512 a^{10} x^{2} \sqrt{a + b x^{2}}}{7436429 b^{6}} + \frac{384 a^{9} x^{4} \sqrt{a + b x^{2}}}{7436429 b^{5}} - \frac{320 a^{8} x^{6} \sqrt{a + b x^{2}}}{7436429 b^{4}} + \frac{40 a^{7} x^{8} \sqrt{a + b x^{2}}}{1062347 b^{3}} - \frac{36 a^{6} x^{10} \sqrt{a + b x^{2}}}{1062347 b^{2}} + \frac{3 a^{5} x^{12} \sqrt{a + b x^{2}}}{96577 b} + \frac{3713 a^{4} x^{14} \sqrt{a + b x^{2}}}{52003} + \frac{12770 a^{3} b x^{16} \sqrt{a + b x^{2}}}{52003} + \frac{990 a^{2} b^{2} x^{18} \sqrt{a + b x^{2}}}{3059} + \frac{31 a b^{3} x^{20} \sqrt{a + b x^{2}}}{161} + \frac{b^{4} x^{22} \sqrt{a + b x^{2}}}{23} & \text{for}\: b \neq 0 \\\frac{a^{\frac{9}{2}} x^{14}}{14} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**13*(b*x**2+a)**(9/2),x)

[Out]

Piecewise((1024*a**11*sqrt(a + b*x**2)/(7436429*b**7) - 512*a**10*x**2*sqrt(a + b*x**2)/(7436429*b**6) + 384*a
**9*x**4*sqrt(a + b*x**2)/(7436429*b**5) - 320*a**8*x**6*sqrt(a + b*x**2)/(7436429*b**4) + 40*a**7*x**8*sqrt(a
+ b*x**2)/(1062347*b**3) - 36*a**6*x**10*sqrt(a + b*x**2)/(1062347*b**2) + 3*a**5*x**12*sqrt(a + b*x**2)/(965
77*b) + 3713*a**4*x**14*sqrt(a + b*x**2)/52003 + 12770*a**3*b*x**16*sqrt(a + b*x**2)/52003 + 990*a**2*b**2*x**
18*sqrt(a + b*x**2)/3059 + 31*a*b**3*x**20*sqrt(a + b*x**2)/161 + b**4*x**22*sqrt(a + b*x**2)/23, Ne(b, 0)), (
a**(9/2)*x**14/14, True))

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Giac [B]  time = 1.8674, size = 879, normalized size = 6.28 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^13*(b*x^2+a)^(9/2),x, algorithm="giac")

[Out]

1/334639305*(7429*(3003*(b*x^2 + a)^(15/2) - 20790*(b*x^2 + a)^(13/2)*a + 61425*(b*x^2 + a)^(11/2)*a^2 - 10010
0*(b*x^2 + a)^(9/2)*a^3 + 96525*(b*x^2 + a)^(7/2)*a^4 - 54054*(b*x^2 + a)^(5/2)*a^5 + 15015*(b*x^2 + a)^(3/2)*
a^6)*a^4/b^6 + 12236*(6435*(b*x^2 + a)^(17/2) - 51051*(b*x^2 + a)^(15/2)*a + 176715*(b*x^2 + a)^(13/2)*a^2 - 3
48075*(b*x^2 + a)^(11/2)*a^3 + 425425*(b*x^2 + a)^(9/2)*a^4 - 328185*(b*x^2 + a)^(7/2)*a^5 + 153153*(b*x^2 + a
)^(5/2)*a^6 - 36465*(b*x^2 + a)^(3/2)*a^7)*a^3/b^6 + 966*(109395*(b*x^2 + a)^(19/2) - 978120*(b*x^2 + a)^(17/2
)*a + 3879876*(b*x^2 + a)^(15/2)*a^2 - 8953560*(b*x^2 + a)^(13/2)*a^3 + 13226850*(b*x^2 + a)^(11/2)*a^4 - 1293
2920*(b*x^2 + a)^(9/2)*a^5 + 8314020*(b*x^2 + a)^(7/2)*a^6 - 3325608*(b*x^2 + a)^(5/2)*a^7 + 692835*(b*x^2 + a
)^(3/2)*a^8)*a^2/b^6 + 276*(230945*(b*x^2 + a)^(21/2) - 2297295*(b*x^2 + a)^(19/2)*a + 10270260*(b*x^2 + a)^(1
7/2)*a^2 - 27159132*(b*x^2 + a)^(15/2)*a^3 + 47006190*(b*x^2 + a)^(13/2)*a^4 - 55552770*(b*x^2 + a)^(11/2)*a^5
+ 45265220*(b*x^2 + a)^(9/2)*a^6 - 24942060*(b*x^2 + a)^(7/2)*a^7 + 8729721*(b*x^2 + a)^(5/2)*a^8 - 1616615*(
b*x^2 + a)^(3/2)*a^9)*a/b^6 + 15*(969969*(b*x^2 + a)^(23/2) - 10623470*(b*x^2 + a)^(21/2)*a + 52837785*(b*x^2
+ a)^(19/2)*a^2 - 157477320*(b*x^2 + a)^(17/2)*a^3 + 312330018*(b*x^2 + a)^(15/2)*a^4 - 432456948*(b*x^2 + a)^
(13/2)*a^5 + 425904570*(b*x^2 + a)^(11/2)*a^6 - 297457160*(b*x^2 + a)^(9/2)*a^7 + 143416845*(b*x^2 + a)^(7/2)*
a^8 - 44618574*(b*x^2 + a)^(5/2)*a^9 + 7436429*(b*x^2 + a)^(3/2)*a^10)/b^6)/b